Computation of entropy and Lyapunov exponent by a shift transform.

We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.

Notable effects of the metal salts on the formation and decay reactions of α-tocopheroxyl radical in acetonitrile solution. The complex formation between α-tocopheroxyl and metal cations.

The measurement of the UV-vis absorption spectrum of α-tocopheroxyl (α-Toc(•)) radical was performed by reacting aroxyl (ArO(•)) radical with α-tocopherol (α-TocH) in acetonitrile solution including four kinds of alkali and alkaline earth metal salts (MX or MX(2)) (LiClO(4), LiI, NaClO(4), and Mg(ClO(4))(2)), using stopped-flow spectrophotometry. The maximum wavelength (λ(max)) of the absorption spectrum of the α-Toc(•) at 425.0 nm increased with increasing concentration of metal salts (0-0.500 M) in acetonitrile, and it approached constant values, suggesting an [α-Toc(•)-M(+) (or M(2+))] complex formation. The stability constants (K) were determined to be 9.2, 2.8, and 45 M(-1) for LiClO(4), NaClO(4), and Mg(ClO(4))(2), respectively. By reacting ArO(•) with α-TocH in acetonitrile, the absorption of ArO(•) disappeared rapidly, while that of α-Toc(•) appeared and then decreased gradually as a result of the bimolecular self-reaction of α-Toc(•) after passing through the maximum. The second-order rate constants (k(s)) obtained for the reaction of α-TocH with ArO(•) increased linearly with an increasing concentration of metal salts. The results indicate that the hydrogen transfer reaction of α-TocH proceeds via an electron transfer intermediate from α-TocH to ArO(•) radicals followed by proton transfer. Both the coordination of metal cations to the one-electron reduced anions of ArO(•) (ArO:(-)) and the coordination of counteranions to the one-electron oxidized cations of α-TocH (α-TocH(•)(+)) may stabilize the intermediate, resulting in the acceleration of electron transfer. A remarkable effect of metal salts on the rate of bimolecular self-reaction (2k(d)) of the α-Toc(•) radical was also observed. The rate constant (2k(d)) decreased rapidly with increasing concentrations of the metal salts. The 2k(d) value decreased at the same concentration of the metal salts in the following order: no metal salt > NaClO(4) > LiClO(4) > Mg(ClO(4))(2). The complex formation between α-Toc(•) and metal cations may stabilize the energy level of the reactants (α-Toc(•) + α-Toc(•)), resulting in the decrease of the rate constant (2k(d)). The alkali and alkaline earth metal salts having a smaller ionic radius of cation and a larger charge of cation gave larger K and k(s) values and a smaller 2k(d) value.

Renormalization group approach to interfacial motion in incompressible Richtmyer-Meshkov instability.

Nonlinear interfacial motion in incompressible Richtmyer-Meshkov instability is theoretically investigated using the renormalization group approach. The amplitude equation describing the asymptotic interfacial motion is derived using this approach. A comparison with calculations carried out by the weakly nonlinear analysis is performed for various Atwood numbers and the validity of the renormalization group approach is discussed. We show that this approach suppresses the divergence in the perturbative solutions obtained by the weakly nonlinear analysis and provides better approximations for the growth rate of bubbles and spikes and interfacial profiles at the asymptotic nonlinear stage without requiring the use of Padé approximants.

Fully nonlinear evolution of a cylindrical vortex sheet in incompressible Richtmyer-Meshkov instability.

Fully nonlinear motion of a circular interface in incompressible Richtmyer-Meshkov instability is investigated by treating it as a nonuniform vortex sheet between two different fluids. There are many features in cylindrical geometry such as the existence of two independent spatial scales, radius and wavelength, and the ingoing and outgoing growth of bubbles and spikes. Geometrical complexities lead to the results that nonlinear dynamics of the vortex sheet is determined from the inward and outward motion rather than bubbles and spikes, and that the nonlinear growth strongly depends on mode number.

Vortex core dynamics and singularity formations in incompressible Richtmyer-Meshkov instability.

Motion of a fluid interface in Richtmyer-Meshkov instability is examined as a vortex sheet with the use of Birkhoff-Rott equation. This equation coupled with an evolution equation of the strength of the vortex sheet can describe all inviscid and incompressible fluid instabilities, i.e., Kelvin-Helmholtz, Rayleigh-Taylor, and Richtmyer-Meshkov instabilities, when Atwood numbers and initial distribution of vorticities are given. With these equations, detailed motion of a vortex core in the Richtmyer-Meshkov instability is investigated. For the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, it is known that the curvature of a vortex sheet diverges at a finite time t=tc. This fact indicates that the solution loses its analyticity at tc. We show that the singularity formation also occurs in the Richtmyer-Meshkov instability and at the same time, accumulation of vorticity to some points where singularities are formed develops to the roll-up of a sheet when the sheet is regularized. We investigate motion of these accumulation points, i.e., vortex cores, and present that their trajectories and the strengths depend on the Atwood numbers.

Analytical and numerical study on a vortex sheet in incompressible Richtmyer-Meshkov instability in cylindrical geometry.

Motion of a fluid interface in the Richtmyer-Meshkov instability in cylindrical geometry is examined analytically and numerically. Nonlinear stability analysis is performed in order to clarify the dependence of growth rates of a bubble and spike on the Atwood number and mode number n involved in the initial perturbations. We discuss differences of weakly and fully nonlinear evolution in cylindrical geometry from that in planar geometry. It is shown that the analytical growth rates coincide well with the numerical ones up to the neighborhood of the break down of numerical computations. Long-time behavior of the fluid interface as a vortex sheet is numerically investigated by using the vortex method and the roll up of the vortex sheet is discussed for different Atwood numbers. The temporal evolution of the curvature of a bubble and spike for several mode numbers is investigated and presented that the curvature of spikes is always larger than that of bubbles. The circulation and the strength of the vortex sheet at the fully nonlinear stage are discussed, and it is shown that their behavior is different for the cases that the inner fluid is heavier than the outer one and vice versa.

Nonlinear evolution of an interface in the Richtmyer-Meshkov instability.

The linear theory of the Richtmyer-Meshkov instability derived by Wouchuk and Nishihara [Phys. Plasmas 4, 3761 (1997)] indicates that the instability is driven by the nonuniform velocity shear left by transmitted and reflected rippled shocks at a corrugated interface. In this work, the nonlinear evolution of the interface has been investigated as a self-interaction of a nonuniform vortex sheet with a density jump. The theory developed shows the importance of the finite density jump and the finite initial corrugation amplitude of the interface. By introducing Lagrangian markers on the interface with proper kinematic boundary conditions, it is shown that stretching and shrinking of the interface occur locally even in the tangential direction. This causes deformation of bubble and spike profiles depending on the Atwood number. The vorticity on the interface for a finite density jump is not conserved in the nonlinear regime. Our results suggest that the spiral structure of the spike is due to local increase and decrease of the vorticity on the interface. Nonlinear analysis shows that the large initial amplitude of the corrugation results in rapid increase of the vorticity, which may also explain the fast roll up motion of the spiral for large amplitudes. With the use of the asymptotic linear growth rate, the nonlinear evolution of the instability is uniquely determined from the initial corrugation amplitude of the interface, the Atwood number, and the incident shock intensity. There is no need to use an impulsive formulation. The analytical nonlinear growth agrees well with the experiment [Dimonte et al., Phys. Plasmas 3, 614 (1996)]. The theory reveals nonlinear properties of the instability, such as the time evolution of the interface profiles and the vorticity on the interface, and also their dependence on the Atwood number and the corrugation amplitude.